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NASA NTRS1989

On the secondary instability in plane Poiseuille flow

Numerical experiments were performed to clarify apparent differences between experimental observations and a theoretical prediction of the secondary instability in plane Poiseuille flow. It is shown that subharmonic breakdown is unlikely in natural transition as a result of the initial growth of what we call the 'minus' modes and consequent forcing of Orr-Sommerfeld modes present in the background noise. Subharmonic breakdown was achieved only when these minus modes were continuously suppressed.

Kim, John↗

NASA NTRS1989

Spectral element simulations of laminar and turbulent flows in complex geometries

Spectral element methods are high-order weighted residual techniques based on spectral expansions of variables and geometry for the Navier-Stokes (NS) and transport equations. Here, practical aspects of these methods and their efficient implementation are examined, and several examples of flows in truly complex geometries are presented. The spectral element discretization for NS equations is introduced, and the convergence of the method is addressed. An efficient data management scheme is discussed in the context of parallel processing computations. The method is validated by comparing the spectral element solutions with the exact eigensolutions for the Orr-Sommerfeld equations in two and three dimensions. Computer-aided flow visualizations are presented for an impulsive flow past a sharp edge wedge. Three-dimensional states of channel flow disrupted by an array of cylindrical eddy promoters are studied, and the results of a direct simulation of the turbulent flow in a plane channel are presented.

Karniadakis, George EM↗

NASA NTRS1990

Theoretical study of turbulent channel flow - Bulk properties, pressure fluctuations, and propagation of electromagnetic waves

In this paper, we apply two theoretical turbulence models, DIA and the recent GISS model, to study properties of a turbulent channel flow. Both models provide a turbulent kinetic energy spectral function E(k) as the solution of a non-linear equation; the two models employ the same source function but different closures. The source function is characterized by a rate n sub s (k) which is derived from the complex eigenvalues of the Orr-Sommerfeld (OS) equation in which the basic flow is taken to be of a Poiseuille type. The O-S equation is solved for a variety of Reynolds numbers corresponding to available experimental data. A physical argument is presented whereby the central line velocity characterizing the basic flow, U0 sup L, is not to be identified with the U0 appearing in the experimental Reynolds number. The theoretical results are compared with two types of experimental data: (1) turbulence bulk properties, and (2) properties that depend strongly on the structure of the turbulence spectrum at low wave numbers. The only existing analytical expression for Pi (k) cannot be used in the present case because it applies to the case of a flat plate, not a finite channel.

Canuto, V. M.↗

NASA NTRS1989

Analysis of turbulence in shear flows using the stabilization principle

An analysis is made based upon the concept that the velocity fluctuations, and therefore, the Reynolds stresses, driven by the instability of the original flow grow until a new stable state is approached. The Reynolds stresses incorporated into the Orr-Sommerfeld equation are coupled with the main flow such that all the imaginary parts of the complex eigenvalues vanish, i.e., the original instability is eliminated. Using this stabilization principle, it is possible to find the Reynolds stresses as well as the mean velocity for plane Poiseuille flow with the Reynolds number slightly higher than the critical.

Zak, M.↗

NASA NTRS1990

Elimination of spurious eigenvalues in the Chebyshev tau spectral method

A very simple modification is presented for the Chebyshev tau method which can eliminate spurious eigenvalues, proceeding from a consideration of the vorticity-streamfunction reformulation of the Chebyshev tau method and the Chebyshev-Galerkin method, which have no spurious modes. Consideration of a model problem indicates that these two approaches are equivalent, and that they reduce to the present modification of the tau method. This modified tau method also eliminates spurious eigenvalues from the Orr-Sommerfeld equation.

Mcfadden, G. B.↗

NASA NTRS1990

Computation of instability and transition

The state of transition calculations in 1989 and desirable future trends are analyzed within the framework of aeronautical applications with emphasis on wall-bounded flows. A shift to studies of the early stages of transition, particularly to receptivity, from studies of the late stages in excessively confined periodic domains is proposed. It is pointed out that early-stage studies require a larger computational domain and more difficult boundary conditions but not a very fine grid. Intermediate strategies, more involved than the classical Orr-Sommerfeld equation but much more economical than full Navier-Stokes equations, are emphasized, along with the need for compressible calculations.

Spalart, P. R.↗

NASA NTRS1990

Numerical simulation of spatially-evolving instability

A computational study of the spatial stability of plane Poiseuille flow is presented. The numerical scheme employs a time-splitting method to integrate the full Navier-Stokes equations using spectral collocation/finite-difference discretization on a non-staggered mesh. The eigenvalue decomposition procedure is applied for the solution of the Poisson equations using the capacitance matrix technique. The buffer domain method is incorporated for the outflow boundary conditions. The input perturbation velocities are obtained by solving the Orr-Sommerfeld equation for the nonlinear eigenvalue problem employing the companion matrix method. Computational results are compared with the linear theory for two-dimensional disturbances.

Danabasoglu, G.↗

NASA NTRS1991

High-resolution threshold photoionization of N2O

Pulsed field ionization (PFI) has been used in conjunction with a coherent VUV source to obtain high-resolution threshold photoelectron spectra for the (000), (010), (020), and (100) vibrational states of the N2O(+) cation. Simulations for the rotational profiles of each vibronic level were obtained by fitting the Buckingham-Orr-Sichel equations using accurate spectroscopic constants for the ground states of the neutral and the ion. The relative branch intensities are interpreted in terms of the partial waves of the outgoing photoelectron to which the ionic core is coupled and in terms of the angular momentum transferred to the core.

Wiedmann, R. T.↗

NASA NTRS1992

Distributed acoustic receptivity in laminar flow control configurations

A model problem related to distributed receptivity to free-stream acoustic waves in laminar flow control (LFC) configurations is studied, within the Orr-Sommerfield framework, by a suitable extension of the Goldstein-Ruban theory for receptivity due to localized disturbances on the airfoil surface. The results, thus, complement the earlier work on the receptivity produced by local variations in the surface suction and/or surface admittance. In particular, we show that the cumulative effect of the distributed receptivity can be substantially larger than that of a single, isolated suction strip or slot. Furthermore, even if the receptivity is spread out over very large distances, the most effective contributions come from a relatively short region in vicinity of the lower branch of the neutral stability curve. The length scale of this region is intermediate to that of the mean of these two length scales. Finally, it is found that the receptivity is effectively dominated by a narrow band of Fourier components from the wall-suction and admittance distributions, roughly corresponding to a detuning of less than ten percent with respect to the neutral instability wavenumber at the frequency under consideration. The results suggest that the drop-off in receptivity magnitudes away from the resonant wavenumber is nearly independent of the frequency parameter.

Choudhari, Meelan↗

NASA NTRS1990

A Numerical Method for Direct Simulation of Turbulence in Complex Geometries

The ultimate goal of this work is to study the flow inside a channel with riblets on one of the two walls. The method has been tested for two dimensional flows in the presence of bodies with a geometrical singularity and for three dimensional flows inside domains described by Cartesian coordinates. The results have been compared with previous numerical simulations and with experimental results. The cases considered are: (1) the growth of Orr-Sommerfield waves in plane Poiseulle flow; (2) the flow over a backward facing step; (3) the flow past a wedge; and (4) the flow inside a narrow channel. Finally, the case of a channel with two large riblets on a wall has been simulated. In this case a limited number of grid points is sufficient, and in spite of the slow convergence for the pressure solver, one is able to obtain solutions with a reasonable amount of computer time. At present solutions with very fine grids in all three directions can not be obtained, due to lack of a fast pressure solver for general curvilinear coordinates.

P Orlandi↗

NASA NTRS1992

Direct numerical simulations of a spatially developing plane wake

In the present paper, direct numerical methods by which to simulate the spatially developing free shear flows in the transitional region are described and the numerical results of a spatially developing plane wake are presented. The incompressible time-dependent Navier-Stokes equations were solved using Pade finite difference approximations in the streamwise direction, a mapped pseudospectral Fourier method in the cross-stream direction, and a third-order compact Runge-Kutta scheme for time advancement. The unstable modes of the Orr-Sommerfeld equations were used to perturb the inlet of the wake. Statistical analyses were performed and some numerical results were compared with experimental measurements. When only the fundamental mode is forced, the energy spectra show amplification of the fundamental and its higher harmonics. In this case, unperturbed alternate vortices develop in the saturation region of the wake. The phase jitter around the fundamental frequency plays a critical role in generating vortices of random shape and spacing. Large- and small-scale distortions of the fundamental structure are observed. Pairing of vortices of the same sign is observed, as well as vortex coupling of vortices of the opposite sign.

Maekawa, Hiroshi↗

NASA NTRS1950

On stability of free laminar boundary layer between parallel streams

An analysis and calculations on the stability of the free laminar boundary layer between parallel streams were made for an incompressible fluid using the Tollmien-Schlichting theory of small disturbances. Because the boundary conditions are at infinity, two solutions of the Orr-Sommerfeld stability equations need not be considered, and the remaining two solutions are exponential in character at the infinite boundaries. The calculations show that the flow is unstable except for very low Reynolds numbers.

Lessen, Martin↗

NASA NTRS1993

Receptivity in parallel flows: An adjoint approach

Linear receptivity studies in parallel flows are aimed at understanding how external forcing couples to the natural unstable motions which a flow can support. The vibrating ribbon problem models the original Schubauer and Skramstad boundary layer experiment and represents the classic boundary layer receptivity problem. The process by which disturbances are initiated in convectively-unstable jets and shear layers has also received attention. Gaster was the first to handle the boundary layer analysis with the recognition that spatial modes, rather than temporal modes, were relevant when studying convectively-unstable flows that are driven by a time-harmonic source. The amplitude of the least stable spatial mode, far downstream of the source, is related to the source strength by a coupling coefficient. The determination of this coefficient is at the heart of this type of linear receptivity study. The first objective of the present study was to determine whether the various wave number derivative factors, appearing in the coupling coefficients for linear receptivity problems, could be reexpressed in a simpler form involving adjoint eigensolutions. Secondly, it was hoped that the general nature of this simplification could be shown; indeed, a rather elegant characterization of the receptivity properties of spatial instabilities does emerge. The analysis is quite distinct from the usual Fourier-inversion procedures, although a detailed knowledge of the spectrum of the Orr-Sommerfeld equation is still required. Since the cylinder wake analysis proved very useful in addressing control considerations, the final objective was to provide a foundation upon which boundary layer control theory may be developed.

Hill, D. Christopher↗

NASA NTRS1993

On the stability of a time dependent boundary layer

The aim of this article is to determine the stability characteristics of a Rayleigh layer, which is known to occur when the fluid above a flat plate has a velocity imparted to it (parallel to the plate). This situation is intrinsically unsteady, however, as a first approximation we consider the instantaneous stability of the flow. The Orr-Sommerfeld equation is found to govern fixed downstream wavelength linear perturbations to the basic flow profile. By the solution of this equation, we can determine the Reynolds numbers at which the flow is neutrally stable; this quasisteady approach is only formally applicable for infinite Reynolds numbers. We shall consider the large Reynolds number limit of the original problem and use a three deck mentality to determine the form of the modes. The results of the two calculations are compared, and the linear large Reynolds number analysis is extended to consider the effect of weak nonlinearity in order to determine whether the system is subcritical or supercritical.

Otto, S. R.↗

NASA NTRS1994

On the nonlinear stability of viscous modes within the Rayleigh problem on an infinite flat plate

The stability has been investigated of the unsteady flow past an infinite flat plate when it is moved impulsively from rest, in its own plane. For small times the instantaneous stability of the flow depends on the linearized equations of motion which reduce in this problem to the Orr-Sommerfeld equation. It is known that the flow for certain values of Reynolds number, frequency and wave number is unstable to Tollmien-Schlichting waves, as in the case of the Blasius boundary layer flow past a flat plate. With increase in time, the unstable waves only undergo growth for a finite time interval, and this growth rate is itself a function of time. The influence of finite amplitude effects is studied by solving the full Navier-Stokes equations. It is found that the stability characteristics are markedly changed both by the consideration of the time evolution of the flow, and by the introduction of finite amplitude effects.

Webb, J. C.↗

NASA NTRS1999

Disturbance Dynamics in Transitional and Turbulent Flows

In order to expand the predictive capability of single-point turbulence closure models to account for the early-stage transition regime, a methodology for the formulation and calibration of model equations for the ensemble-averaged disturbance kinetic energy and energy dissipation rate is presented. First the decay of laminar disturbances and turbulence in mean shear-free flows is studied. In laminar flows, such disturbances are linear superpositions of modes governed by the Orr-Sommerfeld equation. In turbulent flows, disturbances are described through transport equations for representative mean quantities. The link between a description based on a deterministic evolution equation and a probability based mean transport equation is established. Because an uncertainty in initial conditions exists in the laminar as well as the turbulent regime, a probability distribution must be defined even in the laminar case. Using this probability distribution, it is shown that the exponential decay of the linear modes in the laminar regime can be related to a power law decay of both the (ensemble) mean disturbance kinetic energy and the dissipation rate. The evolution of these mean disturbance quantities is then described by transport equations similar to those for the corresponding turbulent decaying flow. Second, homogeneous shear flow, where disturbances can be described by rapid distortion theory (RDT), is studied. The relationship between RDT and linear stability theory is exploited in order to obtain a closed set of modeled equations. The linear disturbance equations are solved directly so that the numerical simulation yields a database from which the closure coefficients in the ensemble-averaged disturbance equations can be determined.

Grosch, Chester E.↗

NASA NTRS2000

Interfacial Stability of Spherically Converging Plasma Jets for Magnetized Target Fusion

A fusion propulsion scheme has been proposed that makes use of the merging of a spherical distribution of plasma jets to dynamically form a gaseous liner to implode a magnetized target to produce the fusion reaction. In this paper, a study is made of the interfacial stability of the interaction of these jets. Specifically, the Orr-Sommerfeld equation is integrated to obtain the growth rate of a perturbation to the primary flow at the interface between the colliding jets. The results lead to an estimate on the tolerances on the relative flow velocities of the merging plasma jets to form a stable, imploding liner. The results show that the maximum temporal growth rate of the perturbed flow at the jet interface is very small in comparison with the time to full compression of the liner. These data suggest that, as far as the stability of the interface between the merging jets is concerned, the formation of the gaseous liner can withstand velocity variation of the order of 10% between the neighboring jets over the density and temperature ranges investigated.

Thio, Y. C. Francis↗

NASA NTRS2001

Interfacial Stability of Converging Plasma Jets for Magnetized Target Fusion

The merging of a spherical distribution of plasma jets to dynamically form a gaseous liner has been proposed for use in magnetized target fusion propulsion. In this paper, a study is made of the interfacial stability of the interaction of these jets. Specifically, the Orr-Sommerfeld equation is integrated to obtain the growth rate of a perturbation to the primary flow at the interface between the colliding jets. The analysis lead to an estimate on the tolerances on the relative flow velocities of the merging plasma jets to form a stable, imploding liner. The results show that during the merging of the jets to form a liner and before contact with the target plasma the growth of the perturbed flow at the jet interface is not likely to destabilize the liner. These data suggest that, as far as the stability of the interface between the merging jets is concerned, the formation of liner can withstand velocity variation up to 50% between the neighboring jets over the density and temperature ranges investigated.

Cassibry, J. T.↗